\(\cot\;\theta ~=~ \dfrac{1}{\tan\;\theta} \qquad \) when \(\tan\;\theta \) is defined and not \(0\)

\(\sin\;\theta ~=~ \dfrac{1}{\csc\;\theta} \qquad \) when \(\csc\;\theta \) is defined and not \(0\)

\(\cos\;\theta ~=~ \dfrac{1}{\sec\;\theta} \qquad \) when \(\sec\;\theta \) is defined and not \(0\)

\(\tan\;\theta ~=~ \dfrac{1}{\cot\;\theta} \qquad \) when \(\cot\;\theta \) is defined and not \(0\)

Notice that each of these equations is true for all angles \(\theta \) for which both sides of the equation are defined. Such equations are called identities, and in this section we will discuss several trigonometric identities, i.e. identities involving the trigonometric functions. These identities are often used to simplify complicated expressions or equations. For example, one of the most useful trigonometric identities is the following:

To prove this identity, pick a point \((x,y) \) on the terminal side of \(\theta \) a distance \(r >0 \) from the origin, and suppose that \(\cos\;\theta \ne 0 \). Then \(x \ne 0 \) (since \(\cos\;\theta = \frac{x}{r}\)), so by definition

Note how we proved the identity by expanding one of its sides (\(\frac{\sin\;\theta}{\cos\;\theta}\)) until we got an expression that was equal to the other side (\(\tan\;\theta\)). This is probably the most common technique for proving identities. Taking reciprocals in the above identity gives:

We will now derive one of the most important trigonometric identities. Let \(\theta \) be any angle with a point \((x,y) \) on its terminal side a distance \(r>0 \) from the origin. By the Pythagorean Theorem, \(r^2 = x^2 + y^2 \) (and hence \(r=\sqrt{x^2 + y^2}\)). For example, if \(\theta \) is in QIII as in Figure 3.1.1, then the legs of the right triangle formed by the reference angle have lengths \(|x| \) and \(|y| \) (we use absolute values because \(x \) and \(y \) are negative in QIII). The same argument holds if \(\theta\) is in the other quadrants or on either axis. Thus,

\[\nonumber
r^2 ~=~ |{x}|^2 ~+~ |{y}|^2 ~=~ x^2 ~+~ y^2 ~,
\]
so dividing both sides of the equation by \(r^2 \) (which we can do since \(r>0\)) gives

You can think of this as sort of a trigonometric variant of the Pythagorean Theorem. Note that we use the notation \(\sin^2 \;\theta \) to mean \((\sin\;\theta)^2 \), likewise for cosine and the other trigonometric functions. We will use the same notation for other powers besides \(2 \).

The above inequalities are not identities (since they are not equations), but they provide useful checks on calculations. Recall that we derived those inequalities from the definitions of sine and cosine in Section 1.4.

In Equation \ref{3.3}, dividing both sides of the identity by \(\cos^2 \;\theta \) gives

In the above example, how did we know to expand the left side instead of the right side? In general, though this technique does not always work, the more complicated side of the identity is likely to be easier to expand. The reason is that, by its complexity, there will be more things that you can do with that expression. For example, if you were asked to prove that

Notice how \(\theta \) does not appear in our final result. The trick was to get a common coefficient (\(ac\)) for \(\;\cos\;\theta\; \) and \(\;\sin\;\theta\; \) so that we could use \(\;\cos^2 \;\theta + \sin^2 \;\theta = 1 \). This is a common technique for eliminating trigonometric functions from systems of equations.